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An introduction to tile-based self-assembly and a survey of recent results

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Abstract

We first give an introduction to the field of tile-based self-assembly, focusing primarily on theoretical models and their algorithmic nature. We start with a description of Winfree’s abstract Tile Assembly Model (aTAM) and survey a series of results in that model, discussing topics such as the shapes which can be built and the computations which can be performed, among many others. Next, we introduce the more experimentally realistic kinetic Tile Assembly Model (kTAM) and provide an overview of kTAM results, focusing especially on the kTAM’s ability to model errors and several results targeted at preventing and correcting errors. We then describe the 2-Handed Assembly Model (2HAM), which allows entire assemblies to combine with each other in pairs (as opposed to the restriction of single-tile addition in the aTAM and kTAM) and doesn’t require a specified seed. We give overviews of a series of 2HAM results, which tend to make use of geometric techniques not applicable in the aTAM. Finally, we discuss and define a wide array of more recently developed models and discuss their various tradeoffs in comparison to the previous models and to each other.

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Notes

  1. The restriction on overlap is a formalization of the physical mechanism of steric protection.

  2. with the convention that \(\infty = \infty + 1 = \infty - 1\)

  3. Note that a supertile \(\tilde{\alpha}\) could be non-terminal in the sense that there is a producible supertile \(\tilde{\beta}\) such that \(C^\tau_{\tilde{\alpha},\tilde{\beta}} \neq \emptyset,\) yet it may not be possible to produce \(\tilde{\alpha}\) and \(\tilde{\beta}\) simultaneously if some tile types are given finite initial counts, implying that \(\tilde{\alpha}\) cannot be “grown” despite being non-terminal. If the count of each tile type in the initial state is ∞, then all producible supertiles are producible from any state, and the concept of terminal becomes synonymous with “not able to grow”, since it would always be possible to use the abundant supply of tiles to assemble \(\tilde{\beta}\) alongside \(\tilde{\alpha}\) and then attach them.

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Acknowledgments

The author would like to thank Scott Summers, Damien Woods, and Dave Doty for much helpful guidance and advice in putting together this survey. He would also like thank Nataša Jonoska and Jérôme Durand-Lose for the invitation to write it. Finally, he is greatly indebted to two anonymous reviewers whose extremely thorough reviews and comments greatly improved this manuscript. This author’s research was supported in part by National Science Foundation Grant CCF-1117672.

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    Matthew J. Patitz

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Patitz, M.J. An introduction to tile-based self-assembly and a survey of recent results. Nat Comput 13, 195–224 (2014). https://doi.org/10.1007/s11047-013-9379-4

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